Parallel pivoting combined with parallel reduction and fill-in control

Parallel algorithms for triangularization of large, sparse, and unsymmetric matrices are presented. The method combines the parallel reduction with a new parallel pivoting technique, control over generation of fill-ins and check for numerical stability, all done in parallel with the work being distributed over the active processes. The parallel pivoting technique uses the compatibility relation between pivots to identify parallel pivot candidates and uses the Markowitz number of pivots to minimize fill-in. This technique is not a preordering of the sparse matrix and is applied dynamically as the decomposition proceeds.